Abstract: The famous question "can you hear the shape of a drum," has inspired quite a bit of mathematical activity due to its rich connections. In this talk, I will describe how this problem has a natural generalization to the study of splitting fields of division algebras and describe some new joint work with Max Lieblich in this direction.
From the point of view of division algebras, a well known result of Amitsur shows that central division algebras which share the same splitting fields must generate the same cyclic subgroup in the Brauer group. On the other extreme, over certain fields, one may find examples of infinite collections of central division algebras which share all the same finite splitting fields. In our work, we show that while one can find infinite collections of central division algebras which share all the same finitely generated splitting fields of transcendence degree one, algebras which share the same finitely generated transcendence degree two splitting fields must generate the same cyclic subgroup of the Brauer group (in characteristic 0).
